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On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types

Abstract : We consider a simplified model for two-phase flows in one-dimensional heterogeneous porous media made of two different rocks. We focus on the effects induced by the discontinuity of the capillarity field at interface. We first consider a model with capillarity forces within the rocks, stating an exis-tence/uniqueness result. Then we look for the asymptotic problem for vanishing capillarity within the rocks, remaining only on the interface. We show that either the solution to the asymptotic problem is the optimal entropy solution to a scalar conservation law with discontinuous flux, or it admits a non-classical shock at the interface modeling oil-trapping. 1. Introduction. We are interested in a simplified model of incompressible im-miscible two-phase flows within heterogeneous porous media made of several rock types. We consider a one-dimensional porous medium —represented by R— made of two porous sub-media —represented by Ω 1 = {x < 0} and Ω 2 = {x > 0}—. For the sake of simplicity, each sub-domain Ω 1 and Ω 2 is supposed to be homogeneous, i.e. its physical properties depend neither on time nor on space. We will focus on the effects of discontinuities arising at the interface between the different rocks, represented in the sequel by {x = 0}. We consider a incompressible immiscible two-phase flow within this medium, driven by gravity/buoyancy forces and by global convection. Such models are particularly used in petrol engineering to predict the motions of oil. The underlying mathematical problem, in the case of homogeneous domains has been widely studied , leading to numerous publications. We refer for example to [4, 6, 15, 21] for detailed informations on such models and their mathematical treatments. The case of domains with " smooth " variations of the data has been studied in [5, 16]. We assume that the fluid is constituted of two immiscible phases, so-called the oil-phase and the water-phase. Considering the conservation of both phases, we obtain the following equation in each Ω i :
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Clément Cancès. On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Networks and Heterogeneous Media, AIMS-American Institute of Mathematical Sciences, 2010. ⟨hal-01713559⟩



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